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A323567 E.g.f. exp( Integral C(x) dx ) = (A(x) + B(x))/7 where A(x) = 3 + Integral B(x)*C(x) dx and B(x) = 4 + Integral A(x)*C(x) dx such that A(x)^2 - 3^2 = B(x)^2 - 4^2 = C(x)^2 - 5^2. 7

%I

%S 1,5,37,430,6857,136940,3273481,91259740,2907969101,104249194220,

%T 4152524574853,181946149653820,8696849043348593,450342763293396860,

%U 25113610370095833889,1500508072578186089260,95630459357796386901749,6475645802688804474935180,464294473578448003768876957,35138647263640258978617313180,2799322643541170608109539354457,234158633483737204648822542925340

%N E.g.f. exp( Integral C(x) dx ) = (A(x) + B(x))/7 where A(x) = 3 + Integral B(x)*C(x) dx and B(x) = 4 + Integral A(x)*C(x) dx such that A(x)^2 - 3^2 = B(x)^2 - 4^2 = C(x)^2 - 5^2.

%H Paul D. Hanna, <a href="/A323567/b323567.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. exp( Integral C(x) dx ) where related series A(x), B(x), and C(x) satisfy the following relations.

%F (1a) A(x) = 3 + Integral B(x)*C(x) dx.

%F (1b) B(x) = 4 + Integral A(x)*C(x) dx.

%F (1c) C(x) = 5 + Integral A(x)*B(x) dx.

%F (2a) C(x)^2 - B(x)^2 = 9.

%F (2b) C(x)^2 - A(x)^2 = 16.

%F (2c) B(x)^2 - A(x)^2 = 7.

%F (3a) A(x)*B(x)*C(x) = A(x)*A'(x) = B(x)*B'(x) = C(x)*C'(x).

%F (3b) Integral 2*A(x)*B(x)*C(x) dx = A(x)^2 - 9 = B(x)^2 - 16 = C(x)^2 - 25.

%F (4a) B(x) + C(x) = 9 * exp( Integral A(x) dx ).

%F (4b) A(x) + C(x) = 8 * exp( Integral B(x) dx ).

%F (4c) A(x) + B(x) = 7 * exp( Integral C(x) dx ).

%e E.g.f. exp( Integral C(x) dx ) = 1 + 5*x + 37*x^2/2! + 430*x^3/3! + 6857*x^4/4! + 136940*x^5/5! + 3273481*x^6/6! + 91259740*x^7/7! + 2907969101*x^8/8! + 104249194220*x^9/9! + 4152524574853*x^10/10! + ...

%e RELATED SERIES.

%e A(x) = 3 + 20*x + 123*x^2/2! + 1540*x^3/3! + 23871*x^4/4! + 480260*x^5/5! + 11449599*x^6/6! + 319491220*x^7/7! + 10176946203*x^8/8! + 364884459380*x^9/9! + 14533663841187*x^10/10! + ... + A323563(n)*x^n/n! + ...

%e such that A(x) = 3 + Integral B(x)*C(x) dx.

%e B(x) = 4 + 15*x + 136*x^2/2! + 1470*x^3/3! + 24128*x^4/4! + 478320*x^5/5! + 11464768*x^6/6! + 319326960*x^7/7! + 10178837504*x^8/8! + 364859900160*x^9/9! + 14534008182784*x^10/10! + ... + A323564(n)*x^n/n! + ...

%e such that B(x) = 4 + Integral A(x)*C(x) dx.

%e C(x) = 5 + 12*x + 125*x^2/2! + 1500*x^3/3! + 24265*x^4/4! + 478236*x^5/5! + 11461625*x^6/6! + 319303500*x^7/7! + 10179006445*x^8/8! + 364862775468*x^9/9! + 14534000631125*x^10/10! + ... + A323565(n)*x^n/n! + ...

%e such that C(x) = 5 + Integral A(x)*B(x) dx.

%e A(x)^2 = 9 + 120*x + 1538*x^2/2! + 24000*x^3/3! + 480400*x^4/4! + 11444160*x^5/5! + 319475984*x^6/6! + 10177152000*x^7/7! + 364886675200*x^8/8! + 14533662074880*x^9/9! + 636813851059712*x^10/10! ...

%e such that C(x)^2 - A(x)^2 = 16 and B(x)^2 - A(x)^2 = 7.

%e A(x) + B(x) = 7 * exp( Integral C(x) dx ) = 7 + 35*x + 259*x^2/2! + 3010*x^3/3! + 47999*x^4/4! + 958580*x^5/5! + 22914367*x^6/6! + 638818180*x^7/7! + 20355783707*x^8/8! + 729744359540*x^9/9! + 29067672023971*x^10/10! + ...

%e A(x) + C(x) = 8 * exp( Integral B(x) dx ) = 8 + 32*x + 248*x^2/2! + 3040*x^3/3! + 48136*x^4/4! + 958496*x^5/5! + 22911224*x^6/6! + 638794720*x^7/7! + 20355952648*x^8/8! + 729747234848*x^9/9! + 29067664472312*x^10/10! + ...

%e B(x) + C(x) = 9 * exp( Integral A(x) dx ) = 9 + 27*x + 261*x^2/2! + 2970*x^3/3! + 48393*x^4/4! + 956556*x^5/5! + 22926393*x^6/6! + 638630460*x^7/7! + 20357843949*x^8/8! + 729722675628*x^9/9! + 29068008813909*x^10/10! + ...

%e exp( Integral A(x) dx ) = 1 + 3*x + 29*x^2/2! + 330*x^3/3! + 5377*x^4/4! + 106284*x^5/5! + 2547377*x^6/6! + 70958940*x^7/7! + 2261982661*x^8/8! + 81080297292*x^9/9! + 3229778757101*x^10/10! + ... + A323569(n)*x^n/n! + ...

%e exp( Integral B(x) dx ) = 1 + 4*x + 31*x^2/2! + 380*x^3/3! + 6017*x^4/4! + 119812*x^5/5! + 2863903*x^6/6! + 79849340*x^7/7! + 2544494081*x^8/8! + 91218404356*x^9/9! + 3633458059039*x^10/10! + ... + A323568(n)*x^n/n! + ...

%e A(x)*B(x)*C(x) = 60 + 769*x + 12000*x^2/2! + 240200*x^3/3! + 5722080*x^4/4! + 159737992*x^5/5! + 5088576000*x^6/6! + 182443337600*x^7/7! + 7266831037440*x^8/8! + 318406925529856*x^9/9! + 15219462171648000*x^10/10! + ... + A323566(n)*x^n/n! + ...

%e such that A(x)*B(x)*C(x) = A(x)*A'(x) = B(x)*B'(x) = C(x)*C'(x).

%o (PARI) {ab7(n) = my(A=3,B=4,C=5); for(i=1,n,

%o A = 3 + intformal(B*C +x*O(x^n));

%o B = 4 + intformal(A*C);

%o C = 5 + intformal(A*B););

%o n! * polcoeff( (A+B)/7 ,n)}

%o for(n=0,30,print1(ab7(n),", "))

%Y Cf. A323563 (A), A323564 (B), A323565 (C), A323566 (A*B*C), A323568 ((A+C)/8), A323569 ((B+C)/9).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 18 2019

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